Abstract

Group selection can overcome individual selection for selfishness and favour altruism if there is variation among the founders of spatially distinct groups, and groups with many altruists become substantially larger (or exist longer) than groups with few. Whether altruism can evolve in populations that do not have an alternation of local population growth and global dispersal ("viscous populations") has been disputed for some time. Limited dispersal protects the altruists from the non-altruists, but also hinders the export of altruism. In this article, we use the Pair Approximation technique (tracking the dynamics of pairs of neighbours instead of single individuals) to derive explicit invasion conditions for rare mutants in populations with limited dispersal. In such viscous populations, invading mutants form clusters, and ultimately, invasion conditions depend on the properties of such clusters. Thus there is selection on a higher level than that of the individual; in fact, invasion conditions define the unit of selection in viscous populations. We treat the evolution of altruism as a specific example, but the method is of more general interest. In particular, an important advantage is that spatial aspects can be incorporated into game theory in a straightforward fashion; we will specify the ESS for a more general model.

The invasion conditions can be interpreted in terms of inclusive fitness. In contrast with Hamilton's model, the coefficient of relatedness is not merely a given genetical constant but depends on local population dynamical processes (birth, dispersal and death of individuals). With a simple birth rate function, Hamilton's rule is recovered: the cost to the donor should be less than the benefit to the recipient weighted with the coefficient of relatedness. As the coefficient of relatedness is roughly inversely proportional to an individual's number of neighbours, benefits to the recipient must be substantial to outweight the costs, confirming earlier studies. We discuss the consequences for the evolution of dispersal and outline how the method may be extended to study evolution in interacting populations. (C) 1998 Academic Press.